Write conclusion

src/thesis/chapters/1_introduction.tex

@@ -3,14 +3,13 @@
 
 % Intro to quantum computing
 
-% TODO: Rephrase
 In 1982, Richard Feynman, motivated by the difficulty of simulating
 quantum-mechanical systems on classical hardware, put forward the
 idea of building computers from quantum hardware themselves
 \cite{feynman_simulating_1982}.
 The use of such quantum computers has since been shown to offer promising
 prospects not only with regard to simulating quantum systems but also
-for solving certain kinds of problems that are classicaly intractable.
+for solving certain kinds of problems that are classically intractable.
 The most prominent example is Shor's algorithm for integer
 factorization \cite{shor_algorithms_1994}.
 
@@ -73,12 +72,12 @@ subsequent decoding process on the measured syndrome.
 
 Another difference between \ac{qec} and classical channel coding is
 the resource constraints.
-For QEC, low latency matters more than low overall computational
+For \ac{qec}, low latency matters more than low overall computational
 complexity, due to the backlog problem
-\cite[Sec.~II.G.3.]{terhal_quantum_2015}: Some gates may turn
+\cite[Sec.~II.G.3.]{terhal_quantum_2015}: Certain gates turn
 single-qubit errors into multi-qubit ones, so errors must be
 corrected beforehand.
-A QEC system that is too slow accumulates a backlog at these points,
+A \ac{qec} system that is too slow accumulates a backlog at these points,
 causing exponential slowdown.
 
 Several code constructions have been proposed for \ac{qec} codes over the years.
@@ -87,43 +86,75 @@ standard for experimental applications for a long time
 \cite[Sec.~I]{koutsioumpas_colour_2025}, due to their
 reliance on only local connections between qubits
 \cite[Sec.~5]{roffe_decoding_2020}.
-Recently, \ac{qldpc} codes have been getting increasingly more
+Recently, \ac{qldpc} codes have been getting increasing
 attention as they have been shown to offer comparable thresholds with
 substantially improved encoding rates \cite[Sec.~1]{bravyi_high-threshold_2024}.
 \ac{qldpc} codes are generally decoded using a syndrome-based variant
 of the \ac{bp} algorithm \cite[Sec.~1]{roffe_decoding_2020}.
+We focus on \ac{qldpc} codes in our work and specifically \ac{bb} codes,
+as they are promising candidates for practical QEC due to their high
+encoding rates, large minimum distances, and short-depth syndrome
+extraction circuits \cite[Sec.~1]{bravyi_high-threshold_2024}.
 
 % DEMs and fault tolerance
 
-\content{Syndrome extraction can also be faulty -> Need for fault tolerance}
-\content{Have to repeat syndrome measurements}
-\content{DEMs one way of implementing fault tolerance: Model more
-error locations -> Larger resulting codes}
-\content{Literature deals with latency problem for fault tolerance by
-sliding-window decoding}
+The syndrome extraction itself is implemented on quantum hardware and
+is therefore subject to the same noise as the data qubits.
+As a consequence, the \ac{qec} procedure, meant to protect the quantum
+state, itself introduces new \emph{internal errors}.
+A procedure is called \emph{fault-tolerant} if it remains effective
+even in the presence of these internal errors
+\cite[Sec.~4]{gottesman_introduction_2009}.
+To deal with internal errors that flip syndrome bits, multiple rounds
+of syndrome measurements are performed.
+
+One approach of implementing fault tolerance is using \acp{dem}.
+A \ac{dem} abstracts away the underlying circuit,
+focusing only on the relationship between possible errors
+and their effects on the syndrome \cite[Sec.~1.4.3]{higgott_practical_2024}.
+A \emph{detector error matrix} is generated from the circuit, which is
+used for decoding instead of the original check matrix.
+Decoding under a \ac{dem} poses a challenge with respect to the
+latency constraint.
+This is because the detector error matrix is much larger than the
+check matrix of the underlying code, since it needs to represent many
+more error locations.
+For example, in our experiments using the $\llbracket 144,12,12
+\rrbracket$ \ac{bb} code with $12$ syndrome measurement rounds, the
+number of \acp{vn} grew from $144$ to $9504$ and the number of
+\acp{cn} grew from $72$ to $1008$.
+
+To keep the latency of \ac{dem} decoding manageable, one approach is
+\emph{sliding-window decoding}.
+Instead of decoding on the entire detector error matrix at once,
+it is partitioned into several overlapping windows.
+Once decoding of one window is complete, error estimates on the initial part
+that is no longer needed are committed, and the next window is processed.
+This way, decoding can start as soon as the syndrome bits required
+for the first window have been extracted.
+The idea originates with the \emph{overlapping recovery} scheme
+proposed for the surface code in
+\cite[Sec.~IV.B]{dennis_topological_2002} and has since been studied
+for surface and toric codes \cite{kuo_fault-tolerant_2024} as well as
+for \ac{qldpc} codes under both phenomenological and circuit-level
+noise \cite{huang_increasing_2024,gong_toward_2024,kang_quits_2025}.
 
 % Reseach gap + our work
 
-\content{Use BP for decoding, but has convergence issues -> Modify BP}
-
-\content{We note a striking similarity between sliding-window
-decoding for DEMs and the way SC-LDPC codes are decoded}
-\content{Extend QEC sliding-window decoding by warm start, inspired
-by SC-LDPC decoders}
-The existing realizations of sliding-window decoding for \ac{qec}
+We observe a structural similarity between sliding-window decoding for
+\acp{dem} and window decoding for \ac{sc}-\acs{ldpc} codes.
+In contrast to the latter, however, where \ac{bp} messages are
+carried between windows \cite[Sec.~III.~C.]{hassan_fully_2016},
+the existing realizations of sliding-window decoding for \ac{qec}
 discard the soft information produced inside one window before moving
-on to the next, in contrast to the analogous \ac{sc}-\ac{ldpc}
-decoders, which carry messages between windows
-\cite[Sec.~III.~C.]{hassan_fully_2016}.
-This thesis investigates whether the same idea can be carried over to
-the \ac{qec} setting.
+to the next.
 We propose \emph{warm-start sliding-window decoding}, in which the
 \ac{bp} messages from the overlap region of the previous window are
 reused to initialize \ac{bp} in the current window in place of the
 standard cold-start initialization.
 We formulate the warm start first for plain \ac{bp} and then for
-\ac{bpgd}, where some care is needed in deciding which information to
-carry over.
+\ac{bpgd}, a variant of \ac{bp} with better convergence properties
+for \ac{qec} codes.
 The decoders are evaluated by Monte Carlo simulation on the
 $\llbracket 144,12,12 \rrbracket$ \ac{bb} code under standard
 circuit-based depolarizing noise over $12$ syndrome extraction rounds.
@@ -131,140 +162,6 @@ The main finding is that warm-starting yields a consistent
 improvement at low iteration budgets, which is the regime relevant for
 low-latency operation.
 
-% The need for fault tolerance
-
-% A naive picture of \ac{qec} treats the syndrome extraction circuit as
-% ideal and only considers errors on the data qubits.
-% In reality, every gate, every ancilla, and every measurement involved
-% in extracting the syndrome can itself fail, introducing new faults
-% into the procedure that is supposed to correct them
-% \cite[Sec.~III]{shor_scheme_1995}.
-% A \ac{qec} procedure is called \emph{fault-tolerant} if it remains
-% effective in the presence of these internal faults
-% \cite[Sec.~4]{gottesman_introduction_2009}.
-
-% Fault tolerance
-
-% The standard formal definition requires the number of output errors
-% to remain bounded as long as the combined number of input and
-% internal errors does not exceed the correction capability of the code
-% \cite[Def.~4.2]{derks_designing_2025}.
-% To deal with internal errors that flip syndrome bits, multiple rounds
-% of syndrome measurements are performed, and the resulting space-time
-% history of detector outcomes is decoded jointly.
-% The probabilities of errors at each location in the circuit are
-% collected in a \emph{noise model}.
-% The most general such model, in which an arbitrary Pauli error is
-% allowed after each gate, is referred to as \emph{circuit-level noise}
-% \cite[Def.~2.5]{derks_designing_2025} and is the noise model that
-% should be used for fault-tolerance simulations
-% \cite[Sec.~4.2]{derks_designing_2025}.
-
-% DEMs
-
-% The combination of circuit-level noise and multiple syndrome
-% measurement rounds yields a complicated, code- and circuit-specific
-% decoding problem.
-% A recent line of work argues that this problem is most cleanly
-% expressed through a \acf{dem} \cite[Sec.~6]{derks_designing_2025}.
-% A \ac{dem} abstracts away the underlying circuit and lists the
-% independent error mechanisms together with the detectors they flip
-% and the logical observables they affect.
-% From the decoder's perspective, decoding under a \ac{dem} is again a
-% classical decoding problem on a parity-check matrix, with the
-% detectors playing the role of \acfp{cn} and the error mechanisms
-% playing the role of \acfp{vn}.
-% The standard tool for generating \acp{dem} from arbitrary stabilizer
-% circuits is Stim \cite{gidney_stim_2021}, in which the \ac{dem}
-% formalism was originally introduced.
-
-% The issues with deocoding under DEMs
-
-% For \ac{qec}, the binding constraint on the decoder is latency, not
-% raw computational complexity.
-% This is the \emph{backlog problem}: certain gates can transform
-% existing single-qubit errors into multi-qubit errors, and any
-% correction must be applied before such gates are reached.
-% A decoder that fails to keep up with the rate at which the hardware
-% produces syndromes leads to an exponential slowdown of the computation
-% \cite[Sec.~II.G.3.]{terhal_quantum_2015}.
-
-% Decoding under a \ac{dem} aggravates this constraint, because the
-% matrix that results from unrolling several rounds of syndrome
-% extraction is much larger than the parity-check matrix of the
-% underlying code.
-% Each error mechanism in the circuit becomes a separate \ac{vn} and
-% each detector becomes a separate \ac{cn}.
-% For the $\llbracket 144,12,12 \rrbracket$ \acf{bb} code
-% \cite[Sec.~3]{bravyi_high-threshold_2024} with $12$ syndrome
-% measurement rounds, the number of \acp{vn} grows from $144$ to $9504$
-% and the number of \acp{cn} grows from $72$ to $1008$.
-
-% Exiting solutions to these issues (sliding-window decoding + BP modifications)
-
-% The dominant strategy for keeping the latency of \ac{dem} decoding
-% manageable is \emph{sliding-window decoding}.
-% Instead of decoding the entire space-time history at once, the
-% decoder operates on a window that spans only a few syndrome
-% measurement rounds.
-% After each round, the window slides forward, and the corrections in
-% the part of the previous window that is no longer needed are committed.
-% The idea originates with the \emph{overlapping recovery} scheme
-% proposed for the surface code in \cite[Sec.~IV.B]{dennis_topological_2002}
-% and has since been studied for surface and toric codes
-% \cite{kuo_fault-tolerant_2024} as well as for \ac{qldpc} codes under
-% both phenomenological and circuit-level noise
-% \cite{huang_increasing_2024,gong_toward_2024,kang_quits_2025}.
-% The structure of the decoding problem inside each window is
-% reminiscent of \acf{sc}-\acf{ldpc} decoding from classical
-% communications \cite[Intro.]{costello_spatially_2014}, where similar
-% windowing techniques are used and where soft information is passed
-% between consecutive windows
-% \cite[Sec.~III.~C.]{hassan_fully_2016}.
-
-% We focus on QLDPC codes
-
-% In this work we focus on \acf{qldpc} codes, of which the \ac{bb} code
-% mentioned above is one example.
-% \ac{qldpc} codes have emerged as leading candidates for practical
-% \ac{qec} due to their high encoding rates and large minimum distances
-% at short syndrome-extraction-circuit depths
-% \cite[Sec.~1]{bravyi_high-threshold_2024}.
-% The natural decoder for them is \acf{bp}, which is well suited to
-% sparse parity-check matrices and admits an efficient and parallel
-% implementation, but is known to converge poorly on quantum codes due
-% to quantum degeneracy and the unavoidable short cycles in the Tanner
-% graph \cite[Sec.~II.C.]{babar_fifteen_2015}\cite[Sec.~V]{roffe_decoding_2020}.
-% Several modifications of \ac{bp} have been proposed to address this:
-% combining \ac{bp} with \acf{osd} \cite{roffe_decoding_2020}, decoding
-% multiple variations of the code in parallel as in \acf{aed}
-% \cite{koutsioumpas_automorphism_2025}, or extending \ac{bp} with
-% guided decimation as in \acf{bpgd} \cite{yao_belief_2024}.
-
-% Contributions of this Thesis
-
-% The existing realizations of sliding-window decoding for \ac{qec}
-% discard the soft information produced inside one window before moving
-% on to the next, in contrast to the analogous \ac{sc}-\ac{ldpc}
-% decoders, which carry messages between windows
-% \cite[Sec.~III.~C.]{hassan_fully_2016}.
-% This thesis investigates whether the same idea can be carried over to
-% the \ac{qec} setting.
-%
-% We propose \emph{warm-start sliding-window decoding}, in which the
-% \ac{bp} messages from the overlap region of the previous window are
-% reused to initialize \ac{bp} in the current window in place of the
-% standard cold-start initialization.
-% We formulate the warm start first for plain \ac{bp} and then for
-% \ac{bpgd}, where some care is needed in deciding which information to
-% carry over.
-% The decoders are evaluated by Monte Carlo simulation on the
-% $\llbracket 144,12,12 \rrbracket$ \ac{bb} code under standard
-% circuit-based depolarizing noise over $12$ syndrome extraction rounds.
-% The main finding is that warm-starting yields a consistent
-% improvement at low iteration budgets, which is the regime relevant for
-% fault-tolerant operation.
-
 % Outline of the Thesis
 
 \Cref{ch:Fundamentals} reviews the fundamentals of classical and
@@ -292,6 +189,7 @@ introduces the proposed warm-start sliding-window decoder for
 plain \ac{bp} and for \ac{bpgd}, and reports numerical results on the
 $\llbracket 144,12,12 \rrbracket$ \ac{bb} code.
 
+% TODO: Possibly extend to mention specific proposed research directions
 \Cref{ch:Conclusion} concludes the thesis and outlines directions for
 further research.
 

src/thesis/chapters/4_decoding_under_dems.tex

@@ -2274,7 +2274,7 @@ In both panels, every curve again exhibits the expected monotonic
 increase of the per-round \ac{ler} with the physical error rate.
 Across both panels and across all parameter choices, the warm-start
 curves lie above the corresponding cold-start curves, i.e.,
-the warm-start variant performsworse than its cold-start counterpart.
+the warm-start variant performs worse than its cold-start counterpart.
 This is the opposite of what we observed for plain \ac{bp}, where
 warm-start improved upon cold-start at every parameter setting.
 The gap between the warm- and cold-start curves additionally widens

src/thesis/chapters/5_conclusion_and_outlook.tex

@@ -1,15 +1,129 @@
 \chapter{Conclusion and Outlook}
 \label{ch:Conclusion}
 
-\content{Takeaway: Warm-start more effective for lower numbers of max
-    iterations (plays into our hands because lower number of iterations
-means lower latency)}
-\content{Warm-start initialization limited to decoding algorithms
-providing relevant soft information}
+% Recap of motivation
 
-\content{\textbf{Ideas for further research}}
-\content{Softer way of decimating VNs}
-\content{Systematic study on using different inner decoders (AED,
-SED, BPGD, ...)}
-\content{Investigate SC-LDPC window decoding wave-like effects}
+This thesis investigated decoding under \acp{dem} for fault-tolerant
+\ac{qec}, with a focus on low-latency decoding methods for \ac{qldpc} codes.
+The repetition of the syndrome measurements, especially under
+consideration of circuit-level noise, leads to a significant increase
+in decoding complexity: in our experiments on the $\llbracket
+144,12,12 \rrbracket$ \ac{bb} code with $12$ syndrome extraction
+rounds, the check matrix grew from 144 \acp{vn} and 72
+\acp{cn} to 9504 \acp{vn} and 1008 \acp{cn}.
+
+% Recap of research gap and own work
+
+Sliding-window decoding addresses the latency constraint by
+exploiting the time-like locality of the syndrome extraction circuit,
+which manifests as a block-diagonal structure in the detector error
+matrix when detectors are defined as the difference of consecutive
+syndrome measurement rounds.
+We drew a comparison to windowed decoding for \ac{sc}-\ac{ldpc}
+codes, but noted that the existing realizations of sliding-window
+decoding discard the soft information produced inside one window
+before moving to the next.
+Building on this observation, we proposed warm-start sliding-window
+decoding, in which the \ac{bp} messages on the edges crossing into
+the overlap region of the previous window are reused to initialise
+the corresponding messages of the next window in place of the
+standard cold-start initialisation.
+
+We formulated the warm start first for plain \ac{bp} and then for
+\ac{bpgd}, the latter being attractive as an inner decoder because it
+addresses the convergence problems caused by short cycles and
+degeneracy in \ac{qldpc} Tanner graphs.
+The decoders were evaluated by Monte Carlo simulation on the
+$\llbracket 144,12,12 \rrbracket$ \ac{bb} code over $12$ syndrome
+extraction rounds under standard circuit-based depolarizing noise.
+We focused on a qualitative analysis, refraining from further
+optimizations such as introducing a normalization parameter for the
+min-sum algorithm.
+
+% Recap of experimental conclusions
+
+For plain min-sum \ac{bp}, the warm start was consistently beneficial
+across the parameter ranges we examined. The size of the gain depended
+on the overlap between consecutive windows: enlarging $W$ or
+shrinking $F$, both of which enlarge the overlap, raised the
+warm-start performance increase.
+We argued that the underlying mechanism is an effective increase in
+the number of \ac{bp} iterations spent on the \acp{vn} in the overlap
+region: each such \ac{vn} is processed by multiple consecutive window
+invocations, and the warm start lets these invocations accumulate
+iterations on the same \acp{vn} rather than restarting from scratch.
+The gain was most pronounced at low numbers of maximum iterations, where
+every additional iteration carries proportionally more information.
+
+For \ac{bpgd}, we noted that more information is available in the
+overlap region of a window: in addition to the \ac{bp} messages,
+there is information about which \acp{vn} were decimated and to what value.
+Passing this decimation information to the next window in addition to
+the messages turned out to worsen the performance considerably, which
+we attributed to a premature hard decision of the \acp{vn} in the
+overlap region.
+Restricting the warm start to the \ac{bp} messages alone, removed this effect.
+The resulting message-only warm start recovered a consistent
+improvement over cold-start that followed the same qualitative
+behaviour as for plain \ac{bp}: larger overlap, achieved by larger
+$W$ or smaller $F$, yielded a larger gain, and the
+performance difference was most pronounced at low numbers of maximum iterations.
+
+% Implications from experimental results
+
+These observations imply that the warm-start modification to
+sliding-window decoding provides a universal improvement, as long as
+some care is taken with specifying the information to be passed to
+the subsequent window.
+Not that this comes at no additional cost to the decoding complexity,
+since the only difference between warm- and cold-start sliding-window
+decoding is the initialization of the \ac{bp} messages.
+We expect similar behavior with other inner decoders that support
+soft information initialization in the overlap region.
+
+% Further research
+
+Several directions for further research emerge from this work.
+The most immediate is an extension of the evaluation to other
+\ac{qldpc} code families, to other circuit-level noise models such as
+SI1000 or EM3, and to a range of code sizes.
+This would clarify the generality of the gain due to the warm-start
+initialization.
+We expect the qualitative findings to carry over, since the
+underlying mechanism is structural rather than code-specific, but
+quantifying the gain across code families and noise models is left to
+future work.
+
+A second direction is a systematic study of inner decoders under the
+warm-start framework.
+We considered plain min-sum \ac{bp} and \ac{bpgd}, but other
+algorithms used for \ac{qldpc} decoding, such as automorphism
+ensemble decoding \cite{koutsioumpas_automorphism_2025} or neural
+\ac{bp} \cite{miao_quaternary_2025} may admit warm-start variants of their own.
+
+A third direction is a softer treatment of the decimation state in \ac{bpgd}.
+Rather than discarding the decimation information of the previous
+window entirely, as in the message-only warm start used here, one
+could encode the decimation decisions as strong but finite biases on
+the channel \acp{llr} of the next window, allowing the new window's parity
+checks to override them if the syndrome calls for it.
+This would interpolate between the two warm-start variants studied here and
+might combine the benefits of both.
+A related question is whether the decimation schedule itself should
+be aware of the window structure, for instance by deferring
+decimation of \acp{vn} in the overlap region until they have been
+visited by the next window.
+
+A final direction is suggested by the structural similarity between
+sliding-window decoding for \acp{dem} and windowed decoding for
+\ac{sc}-\ac{ldpc} codes.
+The current approach to generating the syndrome extraction circuitry
+necessarily leads to a coupling width of one between adjacent
+syndrome measurement rounds.
+A natural question is whether the coupling width could be
+increased, e.g., by interleaving two separate realizations of the
+syndrome measurement circuitry instead of always repeating the same one.
+Work in this direction would also be a step toward bringing
+sliding-window decoding under DEMs within the scope of the analytical
+machinery developed for SC-LDPC codes.